Geometric Measure Theory Read

Title: Uniform distributed measures over R^n II

Date: 02/20/2017

Time: 1:40 PM - 3:00 PM

Place: C517 Wells Hall

Speaker: Shiwen Zhang, MSU

We will continue reading Theorem 3.14 of Preiss's paper [Geometry of measures in R^n: Distribution, rectifiability, and densities]. We will study whether a uniformly distributed measure is flat or curved at infinity. The proof is based on the previous Lemma 3.13 and some basic properties of a symmetric bi-linear form.

Combinatorics and Graph Theory

A graph H is an isometric subgraph of G if d_H(u,v) = d_G(u,v), for every pair u,v in V(H), where d denotes distance. A graph is distance preserving (dp) if it has an isometric subgraph of every possible order. We consider how to add a vertex to a dp graph so that the result is a dp graph. This condition implies that chordal graphs are dp. A graph is sequentially distance preserving (sdp) if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i at least 1. We give an equivalent condition to sequentially distance preserving based upon simplicial orderings. Using this condition, we prove that if a graph does not contain any induced cycles of length 5 or greater, then it is sdp. In closing, we discuss our results, other work and open problems concerning dp graphs.

Mathematical Physics and Gauge Theory

We propose a method, inspired by Free Probability Theory and Random Matrix Theory, that predicts the eigenvalue distribution of quantum many-body systems with generic interactions [1]. At the heart is a 'Slider', which interpolates between two extremes by matching fourth moments. The first extreme treats the non-commuting terms classically and the second treats them 'free'. By 'free' we mean that the eigenvectors are in generic positions. We prove that the interpolation is universal. We then show that free probability theory also captures the density of states of the Anderson model with an arbitrary disorder and with high accuracy [2]. Theory will be illustrated by numerical experiments.
[Joint work with Alan Edelman]
Time permitting we will prove that quantum local Hamiltonians with generic interactions are gapless [3]. In fact, we prove that there is a continuous density of states arbitrary close to the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. We calculate the scaling of the gap with the system's size in the case that the local terms are distributed according to gaussian &#946;&#8722;orthogonal random matrix ensemble.
References:
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[1] Phys. Rev. Lett. 107, 097205 (2011)
[2] Phys. Rev. Lett. 109, 036403 (2012)
[3] R. Movassagh 'Generic Local Hamiltonians are Gapless', (2017)
arXiv:1606.09313v2 [quant-ph]

Geometry and Topology

I will discuss a proof that every finite volume hyperbolic 3-manifold M contains an abundant collection of immersed, $\pi_1$-injective surfaces. These surfaces are abundant in the sense that their lifts to the universal cover separate any pair of disjoint geodesic planes. The proof relies in a major way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary, we recover Wise's theorem that the fundamental group of M is acts properly and cocompactly on a cube complex. This is joint work with Daryl Cooper.

Colloquium

Title: Supercritical Entanglement: counter-example to the area law for quantum matter

Date: 02/23/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Ramis Movassagh, MIT

In recent years, there has been a surge of activities in proposing exactly solvable quantum spin chains with the surprisingly high amount of entanglement entropies (super-logarithmic violations of the area law). We will introduce entanglement and discuss these models. These models have rich connections with combinatorics, random walks, and universality of Brownian excursions. Lastly, we develop techniques for proving the gap and conclude that these models do not have a relativistic conformal field theory description.